
theorem Th45:
  for G1, G2, G3 being _Graph, G4 being DLGraphComplement of G1
  st G1 == G2 & G3 == G4 holds G3 is DLGraphComplement of G2
proof
  let G1, G2, G3 be _Graph, G4 be DLGraphComplement of G1;
  assume A1: G1 == G2 & G3 == G4;
  then the_Vertices_of G4 = the_Vertices_of G3 &
    the_Edges_of G4 = the_Edges_of G3 by GLIB_000:def 34;
  then the_Vertices_of G1 = the_Vertices_of G3 &
    the_Edges_of G3 misses the_Edges_of G1 by Def6;
  then A2: the_Vertices_of G3 = the_Vertices_of G2 &
    the_Edges_of G3 misses the_Edges_of G2 by A1, GLIB_000:def 34;
  A3: G3 is non-Dmulti by A1, GLIB_000:89;
  now
    let v,w be Vertex of G2;
    A4: v is Vertex of G1 & w is Vertex of G1 by A1, GLIB_000:def 34;
    hereby
      given e1 being object such that
        A5: e1 DJoins v,w,G2;
      A6: e1 DJoins v,w,G1 by A1, A5, GLIB_000:88;
      thus not ex e2 being object st e2 DJoins v,w,G3
      proof
        given e2 being object such that
          A7: e2 DJoins v,w,G3;
        e2 DJoins v,w,G4 by A1, A7, GLIB_000:88;
        hence contradiction by A4, A6, Def6;
      end;
    end;
    assume A8: not ex e2 being object st e2 DJoins v,w,G3;
    not ex e2 being object st e2 DJoins v,w,G4
    proof
      given e2 being object such that
        A9: e2 DJoins v,w,G4;
      e2 DJoins v,w,G3 by A1, A9, GLIB_000:88;
      hence contradiction by A8;
    end;
    then consider e1 being object such that
      A10: e1 DJoins v,w,G1 by A4, Def6;
    take e1;
    thus e1 DJoins v,w,G2 by A1, A10, GLIB_000:88;
  end;
  hence thesis by A2, A3, Def6;
end;
